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The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form p/q, what can you say about the prime factors of q? 

i. 43.123456789 

ii. 0.120120012000120000… 

iii. 43.123456789

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Solution:
i. Since the given number has a terminating decimal expansion, it is a rational number of the form p/q and q is of the form 2m x 5n.
The prime factors of p and q will be either 2 or 5 or both.

ii. Since the decimal expansion is neither terminating nor recurring, the given number is an irrational number.
iii. Since the decimal expansion is nonterminating repeating, the given number is a
rational number of the form p/q and q is not of the form 2m x 5n.
The prime factors of q will also have a factor other than 2 or 5.

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